Details

If the center of the circle is at the origin, the image is not an airfoil but a line segment. For all other choices of center, the circle passes through one point at which the mapping fails to be conformal and encloses the other. This means the mapping is conformal everywhere in the exterior of the circle, so we can model the airflow across an cylinder using a complex analytic potential and then conformally transform to the airflow across an airfoil. The fact that the circle passes through exactly one of these two points means that the image has exactly one cusp and is smooth everywhere else.